This textbook introduces elliptic curves and modular forms, focusing on their connections to number theory, particularly the congruent number problem. It aims to make the subject accessible to those less familiar with advanced mathematics, featuring practical examples and exercises. It's suitable for graduate students and advanced undergraduates in related fields.
This book is a comprehensive introduction to p-adic number theory, now enhanced with a detailed exploration of p-adic functions, including the Iwasawa logarithm and p-adic gamma-function. It features new exercises and an appendix with solutions. Neal Koblitz, an expert in the field, provides valuable insights from his extensive research and teaching experience.
Focusing on algebraic methods in cryptography, this textbook is tailored for graduate and advanced undergraduate students, as well as scientists from other disciplines. It begins with an intuitive introduction to fundamental concepts and techniques, particularly emphasizing computational complexity in a practical context. Unique to this book are discussions on emerging cryptographic systems, including hidden monomial, combinatorial-algebraic, and hyperelliptic systems. With numerous exercises and answers, it serves well for coursework or independent study in mathematics or computer science.
Focusing on p-adic analysis and number theory, this work presents recent advancements made by mathematicians over the past five years. It begins with foundational concepts, including the construction of p-adic numbers and power series, before delving into the properties of p-adic Dirichlet L-series through p-adic measures. The text introduces p-adic gamma functions and examines their connections to L-series, emphasizing parallels with complex analysis. Additionally, it proves a formula for Gauss sums related to p-adic gamma functions, making it valuable for graduate students and researchers in relevant fields.
The book explores the evolving relevance of number theory, particularly in the realms of cryptography and information transmission, contrasting G. H. Hardy's views on its remoteness from practical applications. It highlights the rise of computational number theory, driven by advancements in computer technology. Aimed at readers with little background in algebra or number theory, it introduces key arithmetic topics—both historical and contemporary—using an algorithmic approach to emphasize the efficiency of various techniques.
Książka ta jest poświęcona związkom teorii liczb z kryptografią. Do zrozumienia omawianych w niej problemów nie jest właściwie potrzebna znajomość algebry i teorii liczb. Można tu bowiem znaleźć definicje wszystkich niezbędnych pojęć i dowody wielu prostych twierdzeń. Wykład jest prowadzony w przystępny i interesujący sposób, z wieloma trafnie dobranymi przykładami. Na końcu każdego rozdziału znajdują się ćwiczenia umożliwiające sprawdzenie nabytych wiadomości, a na końcu książki - poprawne do nich odpowiedzi.